Editing Fourier inversion theorem (section)

===Square integrable functions===

In this case the Fourier transform cannot be defined directly as an integral since it may not be absolutely convergent, so it is instead defined by a density argument (see the [[Fourier_transform#On_Lp_spaces|Fourier transform article]]). For example, putting 
:<math>g_k(\xi):=\int_{\{y\in\mathbb{R}^n:\left\vert y\right\vert\leq k\}} e^{-2\pi iy\cdot\xi} \, f(y)\,dy,\qquad k\in\mathbb{N},</math>
we can set <math>\textstyle\mathcal{F}f := \lim_{k\to\infty}g_k</math> where the limit is taken in the <math>L^2</math>-norm. The inverse transform may be defined by density in the same way or by defining it in terms of the Fourier transform and the flip operator. We then have

:<math>f(x)=\mathcal{F}(\mathcal{F}^{-1}f)(x)=\mathcal{F}^{-1}(\mathcal{F}f)(x)</math> 

in the [[Lp space|mean squared norm]]. In one dimension (and one dimension only), it can also be shown that it converges for [[almost every]] {{math|''x''∈ℝ}}- this is [[Carleson's theorem]], but is much harder to prove than convergence in the mean squared norm.